1 /* $OpenBSD: e_log2l.c,v 1.1 2011/07/06 00:02:42 martynas Exp $ */
2
3 /*
4 * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
5 *
6 * Permission to use, copy, modify, and distribute this software for any
7 * purpose with or without fee is hereby granted, provided that the above
8 * copyright notice and this permission notice appear in all copies.
9 *
10 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
11 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
12 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
13 * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
14 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
15 * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
16 * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
17 */
18
19 /* log2l.c
20 * Base 2 logarithm, 128-bit long double precision
21 *
22 *
23 *
24 * SYNOPSIS:
25 *
26 * long double x, y, log2l();
27 *
28 * y = log2l( x );
29 *
30 *
31 *
32 * DESCRIPTION:
33 *
34 * Returns the base 2 logarithm of x.
35 *
36 * The argument is separated into its exponent and fractional
37 * parts. If the exponent is between -1 and +1, the (natural)
38 * logarithm of the fraction is approximated by
39 *
40 * log(1+x) = x - 0.5 x^2 + x^3 P(x)/Q(x).
41 *
42 * Otherwise, setting z = 2(x-1)/x+1),
43 *
44 * log(x) = z + z^3 P(z)/Q(z).
45 *
46 *
47 *
48 * ACCURACY:
49 *
50 * Relative error:
51 * arithmetic domain # trials peak rms
52 * IEEE 0.5, 2.0 100,000 2.6e-34 4.9e-35
53 * IEEE exp(+-10000) 100,000 9.6e-35 4.0e-35
54 *
55 * In the tests over the interval exp(+-10000), the logarithms
56 * of the random arguments were uniformly distributed over
57 * [-10000, +10000].
58 *
59 */
60
61
62
63 /* Coefficients for ln(1+x) = x - x**2/2 + x**3 P(x)/Q(x)
64 * 1/sqrt(2) <= x < sqrt(2)
65 * Theoretical peak relative error = 5.3e-37,
66 * relative peak error spread = 2.3e-14
67 */
68 static const long double P[13] =
69 {
70 1.313572404063446165910279910527789794488E4L,
71 7.771154681358524243729929227226708890930E4L,
72 2.014652742082537582487669938141683759923E5L,
73 3.007007295140399532324943111654767187848E5L,
74 2.854829159639697837788887080758954924001E5L,
75 1.797628303815655343403735250238293741397E5L,
76 7.594356839258970405033155585486712125861E4L,
77 2.128857716871515081352991964243375186031E4L,
78 3.824952356185897735160588078446136783779E3L,
79 4.114517881637811823002128927449878962058E2L,
80 2.321125933898420063925789532045674660756E1L,
81 4.998469661968096229986658302195402690910E-1L,
82 1.538612243596254322971797716843006400388E-6L
83 };
84 static const long double Q[12] =
85 {
86 3.940717212190338497730839731583397586124E4L,
87 2.626900195321832660448791748036714883242E5L,
88 7.777690340007566932935753241556479363645E5L,
89 1.347518538384329112529391120390701166528E6L,
90 1.514882452993549494932585972882995548426E6L,
91 1.158019977462989115839826904108208787040E6L,
92 6.132189329546557743179177159925690841200E5L,
93 2.248234257620569139969141618556349415120E5L,
94 5.605842085972455027590989944010492125825E4L,
95 9.147150349299596453976674231612674085381E3L,
96 9.104928120962988414618126155557301584078E2L,
97 4.839208193348159620282142911143429644326E1L
98 /* 1.000000000000000000000000000000000000000E0L, */
99 };
100
101 /* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2),
102 * where z = 2(x-1)/(x+1)
103 * 1/sqrt(2) <= x < sqrt(2)
104 * Theoretical peak relative error = 1.1e-35,
105 * relative peak error spread 1.1e-9
106 */
107 static const long double R[6] =
108 {
109 1.418134209872192732479751274970992665513E5L,
110 -8.977257995689735303686582344659576526998E4L,
111 2.048819892795278657810231591630928516206E4L,
112 -2.024301798136027039250415126250455056397E3L,
113 8.057002716646055371965756206836056074715E1L,
114 -8.828896441624934385266096344596648080902E-1L
115 };
116 static const long double S[6] =
117 {
118 1.701761051846631278975701529965589676574E6L,
119 -1.332535117259762928288745111081235577029E6L,
120 4.001557694070773974936904547424676279307E5L,
121 -5.748542087379434595104154610899551484314E4L,
122 3.998526750980007367835804959888064681098E3L,
123 -1.186359407982897997337150403816839480438E2L
124 /* 1.000000000000000000000000000000000000000E0L, */
125 };
126
127 static const long double
128 /* log2(e) - 1 */
129 LOG2EA = 4.4269504088896340735992468100189213742664595E-1L,
130 /* sqrt(2)/2 */
131 SQRTH = 7.071067811865475244008443621048490392848359E-1L;
132
133
134 /* Evaluate P[n] x^n + P[n-1] x^(n-1) + ... + P[0] */
135
136 static long double
neval(long double x,const long double * p,int n)137 neval (long double x, const long double *p, int n)
138 {
139 long double y;
140
141 p += n;
142 y = *p--;
143 do
144 {
145 y = y * x + *p--;
146 }
147 while (--n > 0);
148 return y;
149 }
150
151
152 /* Evaluate x^n+1 + P[n] x^(n) + P[n-1] x^(n-1) + ... + P[0] */
153
154 static long double
deval(long double x,const long double * p,int n)155 deval (long double x, const long double *p, int n)
156 {
157 long double y;
158
159 p += n;
160 y = x + *p--;
161 do
162 {
163 y = y * x + *p--;
164 }
165 while (--n > 0);
166 return y;
167 }
168
169
170
171 long double
log2l(long double x)172 log2l(long double x)
173 {
174 long double z;
175 long double y;
176 int e;
177 int64_t hx, lx;
178
179 /* Test for domain */
180 GET_LDOUBLE_WORDS64 (hx, lx, x);
181 if (((hx & 0x7fffffffffffffffLL) | lx) == 0)
182 return __math_divzerol(1);
183 if (hx < 0)
184 return __math_invalidl(x);
185 if (hx >= 0x7fff000000000000LL)
186 return (x + x);
187
188 /* separate mantissa from exponent */
189
190 /* Note, frexp is used so that denormal numbers
191 * will be handled properly.
192 */
193 x = frexpl (x, &e);
194
195
196 /* logarithm using log(x) = z + z**3 P(z)/Q(z),
197 * where z = 2(x-1)/x+1)
198 */
199 if ((e > 2) || (e < -2))
200 {
201 if (x < SQRTH)
202 { /* 2( 2x-1 )/( 2x+1 ) */
203 e -= 1;
204 z = x - 0.5L;
205 y = 0.5L * z + 0.5L;
206 }
207 else
208 { /* 2 (x-1)/(x+1) */
209 z = x - 0.5L;
210 z -= 0.5L;
211 y = 0.5L * x + 0.5L;
212 }
213 x = z / y;
214 z = x * x;
215 y = x * (z * neval (z, R, 5) / deval (z, S, 5));
216 goto done;
217 }
218
219
220 /* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */
221
222 if (x < SQRTH)
223 {
224 e -= 1;
225 x = 2.0L * x - 1.0L; /* 2x - 1 */
226 }
227 else
228 {
229 x = x - 1.0L;
230 }
231 z = x * x;
232 y = x * (z * neval (x, P, 12) / deval (x, Q, 11));
233 y = y - 0.5L * z;
234
235 done:
236
237 /* Multiply log of fraction by log2(e)
238 * and base 2 exponent by 1
239 */
240 z = y * LOG2EA;
241 z += x * LOG2EA;
242 z += y;
243 z += x;
244 z += e;
245 return (z);
246 }
247
